How much percentage are the Pythagorean triples among numbers in a certain population of numbers? Say in 100, 1000 etc like that is calculated in prime number theorem.
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$\begingroup$ I am inclined to agree with Arthur. Maybe, you could rephrase your question to read how many triples (a,b,c) are there where a,b,c < n? That looks like a straight computation problem although I have never seen a formula for any n. $\endgroup$ – bobbym Mar 14 '14 at 15:49
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1$\begingroup$ It would seem that the question is asking how common Pythagorean triples are in $\mathbb{N}^3$. So, what is the probability that a random triple is a Pythagorean triple? $\endgroup$ – N. Owad Mar 14 '14 at 16:37
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$\begingroup$ @daniel Your comment slightly confused me. If the point $(x,y)$ is on the unit circle, it would have hypotenuse one. So you must mean that you want to scale by $c$ so that $cx$ and $cy$ are integers. Then either $c$ is a perfect square or not. And you have to find the rational points of the circle, which is bothersome, to me. $\endgroup$ – N. Owad Mar 14 '14 at 19:36
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According to the OEIS, the result is by Lehmer:
https://oeis.org/search?q=A101931&language=english&go=Search
Then number of primitive triples a_n with the hypotenuse less than 10^n is
$$ \frac{a_n}{10^n} = \frac{1}{2\pi}$$
as n approches infinity.
For the number of Pythagorean triples with hypotenuse < 10^n
https://oeis.org/search?q=A101929&sort=&language=english&go=Search
no closed form is given. Meaning that it is probably unknown.
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1$\begingroup$ Discussion of this point at p. 38 of Lehmer's paper. $\endgroup$ – daniel Mar 14 '14 at 16:59
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First of all, a Pythagorean triple is a triple, so asking how dense they are on the number line doesn't really make sense. That being said, are you familiar with the way of describing any Pythagorean triple? Choose any numbers $m, n$. Then $$ (m^2 - n^2)^2 + (2mn)^2 = (m^2 + n^2)^2 $$ So any number that is either the sum of two squares, the difference between two squares, or a composite, even number (those are darn common, I tell you) is part of a Pythagorean triple. If you're looking for primitive triples, then it's a bit more difficult.
